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Numbers n such that the average of the squares of the proper divisors of n is an integer.
1

%I #10 Jul 26 2019 17:00:57

%S 2,3,5,7,8,9,10,11,13,14,17,19,22,23,25,26,29,31,34,35,37,38,41,43,46,

%T 47,49,50,53,55,58,59,61,62,65,67,71,73,74,75,77,79,80,81,82,83,85,86,

%U 89,91,94,95,97,101,103,106,107,109,113,115,118,119,121,122,125,127,131,133,134,137,139,140,142,143,145,146,149

%N Numbers n such that the average of the squares of the proper divisors of n is an integer.

%C Numbers n such that number of proper divisors of n (A032741) divides sum of squares of proper divisors of n (A067558).

%C All the prime numbers are in this sequence.

%H Charles R Greathouse IV, <a href="/A279814/b279814.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>

%e 8 is in the sequence because 8 has 3 proper divisors {1,2,4}, 1^2 + 2^2 + 4^2 = 21 and 3 divides 21.

%t Select[Range[150], Mod[DivisorSigma[2, #1] - #1^2, DivisorSigma[0, #1] - 1] == 0 &]

%t Select[Range[200],IntegerQ[Mean[Most[Divisors[#]]^2]]&] (* _Harvey P. Dale_, Jul 26 2019 *)

%o (PARI) is(n)=my(d=divisors(n)); d=apply(k->k^2, d[1..#d-1]); n>1 && sum(i=1,#d,d[i])%#d==0 \\ _Charles R Greathouse IV_, Dec 19 2016

%Y Cf. A020486, A023884, A032741, A067558.

%K nonn,easy

%O 1,1

%A _Ilya Gutkovskiy_, Dec 19 2016